How can I read and parse text files with configurable contents?

Question 1

Yes is a good idea, use Newtonsoft.Json to help you with the config load like

private class StartSettings
{
    public string CityReg;
    public int CityNum;
    public string ZipReg;
    public int ZipNum;
    public string StreetReg;
    public int StreetNum;
    public string NumberReg;
    public int NumberNum;
}

var configString = File.ReadAllText(configFilePath);
var config = JsonConvert.DeserializeObject<StartSettings>(configString);

And to read the files just use

Regex rgxStreet = new Regex(config.StreetReg);
Regex rgxNumber = new Regex(config.NumberReg);
Regex rgxCity = new Regex(config.CityReg);
Regex rgxZIP = new Regex(config.ZipReg);

foreach (var line = File.ReadLines(fileName, Encoding.GetEncoding(65001))
                        .Select(l => l.TrimEnd(';').Trim())
{
    if(config.CityNum == i && rgxCity.IsMatch(line))
        a.City = line;
    ...
    i++;
}
return a;

Question 2

Since I doubt it is possible to determine if a value is a street or Cityname, you need to specifiy atleast some information on iput-data in what "format" the data is made up.

If it is possible for you to still decide dataformat go for XML.

Use XML and XmlSerializer like so:

[Serializable]
public class AdressData
{
    [XmlArrayItem("Adress")]
    public Adress[] Adresses

}

[Serializable]
public class Adress
{
    public string Street {get; set;}
    public int Number {get; set;}
    public int Zip{get; set;}
    public string City{get; set;}
    public string State{get; set;}
}

Then use it like this:

XmlSerializer serializer = new XmlSerializer(typeof(AdressData));
AdressData data = (AdressData)serializer.Deserialize(File.Open(fileName));

foreach(Adress adress in data.Adresses)
{
    checkIfItExists(adress);
}

Your XMl should look like this:

<AdressData>
  <Adresses>
    <Adress>
         <Street>WhateverStr</Street>
         <Number>7</Number>
         <Zip>5675765</Zip>
         <City>Citytown</City>
         <State>Alabama</State>
    </Adress>
      <Adress>
         <!-- Order doesnt matter here -->
         <Number>7</Number>
         <Zip>5675765</Zip>
         <City>Citytown</City>
         <State>Alabama</State>
         <Street>WhateverStr</Street>
    </Adress>
  </Adresses>
</AdressData>

The order of the data in the XML doesnt matter, as long as it fitts in the hirearchy. The serializer does some Validation e.g. tries to parse numeric values. All you need to do is check whether the information itself is valid.

It is capable of parsing Enums aswell, so you could (wouldnt recommend though) create an Enum containing all US-Statenames...